Angular Displacement Chapter 7 Circular motion about AXIS Three measures of angles 1 Degrees Rotational Motion Universal Law of Gravitation Kepler s Laws Angular Displacement cont 3 Radians 2 rad s 360 deg Example 7 1 An automobile wheel has a radius of 42 cm If a car drives 10 km through what angle has the wheel rotated Change in distance of a point s 2 r N N counts revolutions r is in radians Angular Speed 2 Revolutions 1 rev 360 deg a In revolutions Can be given in Revolutions s Radians s Called Linear f i t in radians Speed at r N v 2 r revolutions t 2 r f i in rad s 2 t v r a N 3789 b In radians b 2 38x104 radians c In degrees c 1 36x106 degrees Example 7 2 A race car engine can turn at a maximum rate of 12 000 rpm revolutions per minute a What is the angular velocity in radians per second b If helipcopter blades were attached to the crankshaft while it turns with this angular velocity what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound DATA The speed of sound is 343 m s a 1256 rad s b 27 cm Angular Acceleration Denoted by Rotational Linear Equivalence x 0 v0 f vf f i t must be in radians per sec Units are rad s Every point on rigid object has same and Linear and Rotational Motion Analogies Rotational Motion Linear Motion x 0 f t 2 f 0 t v0 v f t 2 1 x v0 t at 2 2 1 x v f t at 2 2 2f v 2f 2 2 2 v02 2 Distance Speed v r A pottery wheel is accelerated uniformly from rest to a rate of 10 rpm in 30 seconds a What was the angular acceleration in rad s2 b How many revolutions did the wheel undergo during that time a 0 0349 rad s2 b 2 50 revolutions a x Linear movement of a rotating point x r Example 7 3 v f v0 at 1 0 t t 2 2 1 2 f t t 2 02 a t t Different points have different linear speeds Example 7 4 A coin of radius 1 5 cm is initially rolling with a rotational speed of 3 0 radians per second and comes to a rest after experiencing a slowing down of 0 05 rad s2 a Over what angle in radians did the coin rotate Acceleration a r Only works for angles in radians b What linear distance did the coin move a 90 rad b 135 cm Centripetal Acceleration Centripetal Acceleration cont Moving in circle at constant SPEED does not mean constant VELOCITY Centripetal acceleration results from CHANGING DIRECTION of the velocity Acceleration directed toward center of circle v a t Derivation a 2r v2 r From the geometry of the Figure Forces Causing Centripetal Acceleration v 2vsin 2 v for small Newton s Second Law F ma From the definition of angular velocity Radial acceleration requires radial force Examples of forces Spinning ball on a string Gravity Electric forces e g atoms v v a t t v a v 2 r Example 7 5a An astronaut is in cirular orbit around the Earth Which vector might describe the astronaut s velocity a Vector A b Vector B c Vector C v2 r A B C Example 7 5b An astronaut is in cirular orbit around the Earth Which vector might describe the astronaut s acceleration a Vector A b Vector B c Vector C A B C A Example 7 5c Example 7 6a B An astronaut is in cirular orbit around the Earth C Dale Earnhart drives 150 mph around a circular track at constant speed A C Neglecting air resistance which vector best describes the frictional force exerted on the tires from contact with the pavement a Vector A b Vector B c Vector C Which vector might describe the gravitional force acting on the astronaut a Vector A b Vector B c Vector C Example 7 6b Dale Earnhart drives 150 mph around a circular track at constant speed B B A Ball on String Demo C Which vector best describes the frictional force Dale Earnhart experiences from the seat a Vector A b Vector B c Vector C Example 7 7 A space station is constructed like a barbell with two 1000 kg compartments separated by 50 meters that spin in a circle r 25 m The compartments spin once every 10 seconds a What is the acceleration at the extreme end of the compartment Give answer in terms of g s b If the two compartments are held together by a cable what is the tension in the cable a 9 87 m s2 1 01 g s b 9870 N DEMO FLYING POKER CHIPS Example 7 8 Example 7 9 A race car speeds around a circular track a If the coefficient of friction with the tires is 1 1 what is the maximum centripetal acceleration in g s that the race car can experience b What is the minimum circumference of the track that would permit the race car to travel at 300 km hr a 1 1 g s b 4 04 km in real life curves are banked A curve with a radius of curvature of 0 5 km on a highway is banked at an angle of 20 If the highway were frictionless at what speed could a car drive without sliding off the road 42 3 m s 94 5 mph Skip Example 7 10 Example 7 11a AAyo yo is spun in a circle as shown If the length of the string is L 35 cm and the circular path is repeated 1 5 times per second at what angle with respect to the vertical does the string bend Which vector represents acceleration 71 6 degrees a A b E c F d B e J Example 7 11c Example 7 11b Which vector represents net force acting on car If car moves at design speed which vector represents the force acting on car from contact with road a A b E a D b E c F d B c G d I e J e J Example Example 7 12 skip A roller coaster goes upside down performing a circular loop of radius 15 m What speed does the roller coaster need at the top of the loop so that it does not need to be held onto the track If car moves slower than design speed which vector represents frictional force acting on car from contact with road neglect air resistance a B b C c E d F 12 1 m s e I Accelerating Reference Frames Consider a frame that is accelerating with af F ma F ma f m a a f Fictitious force Looks like gravitational force If frame acceleration g fictitious force cancels real gravity Examples Falling elevator planetary orbit rotating space stations Newton s …
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